LMFDB - Embedded newform 9300.2.g.q.3349.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2611456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 16x^{2} - 8x + 2 $ x^6 - 2x^3 + 16x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.4
Root			$0.258652 - 0.258652i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.q.3349.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.34889i q^{7} -1.00000 q^{9} +4.69779 q^{11} -2.38350i q^{13} +1.21509i q^{17} -1.03461 q^{19} +1.34889 q^{21} +6.24970i q^{23} -1.00000i q^{27} +5.59859 q^{29} +1.00000 q^{31} +4.69779i q^{33} -5.41811i q^{37} +2.38350 q^{39} +1.30221 q^{41} +5.73240i q^{43} -3.48270i q^{47} +5.18048 q^{49} -1.21509 q^{51} +2.51730i q^{53} -1.03461i q^{57} -7.86620 q^{59} -5.46479 q^{61} +1.34889i q^{63} +1.41811i q^{67} -6.24970 q^{69} +3.79698 q^{71} -1.41811i q^{73} -6.33682i q^{77} +1.81952 q^{79} +1.00000 q^{81} +13.2151i q^{83} +5.59859i q^{87} +2.20302 q^{89} -3.21509 q^{91} +1.00000i q^{93} -6.69779i q^{97} -4.69779 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.34889i	− 0.509834i	−0.966963	−	0.254917i	$-0.917952\pi$
$7$	− 1.34889i	− 0.509834i	0.966963	−	0.254917i	$-0.0820482\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	4.69779	1.41644	0.708218	−	0.705994i	$-0.249499\pi$
$11$	4.69779	1.41644	0.708218	+	0.705994i	$0.249499\pi$
$12$	0	0
$13$	− 2.38350i	− 0.661065i	−0.943795	−	0.330532i	$-0.892772\pi$
$13$	− 2.38350i	− 0.661065i	0.943795	−	0.330532i	$-0.107228\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.21509i	0.294703i	0.989084	+	0.147352i	$0.0470749\pi$
$17$	1.21509i	0.294703i	−0.989084	+	0.147352i	$0.952925\pi$
$18$	0	0
$19$	−1.03461	−0.237355	−0.118678	−	0.992933i	$-0.537866\pi$
$19$	−1.03461	−0.237355	−0.118678	+	0.992933i	$0.537866\pi$
$20$	0	0
$21$	1.34889	0.294353
$22$	0	0
$23$	6.24970i	1.30315i	0.758583	+	0.651576i	$0.225892\pi$
$23$	6.24970i	1.30315i	−0.758583	+	0.651576i	$0.774108\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	5.59859	1.03963	0.519816	−	0.854278i	$-0.326000\pi$
$29$	5.59859	1.03963	0.519816	+	0.854278i	$0.326000\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	4.69779i	0.817780i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 5.41811i	− 0.890732i	−0.895349	−	0.445366i	$-0.853074\pi$
$37$	− 5.41811i	− 0.890732i	0.895349	−	0.445366i	$-0.146926\pi$
$38$	0	0
$39$	2.38350	0.381666
$40$	0	0
$41$	1.30221	0.203371	0.101686	−	0.994817i	$-0.467576\pi$
$41$	1.30221	0.203371	0.101686	+	0.994817i	$0.467576\pi$
$42$	0	0
$43$	5.73240i	0.874182i	0.899417	+	0.437091i	$0.143991\pi$
$43$	5.73240i	0.874182i	−0.899417	+	0.437091i	$0.856009\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3.48270i	− 0.508003i	−0.967204	−	0.254002i	$-0.918253\pi$
$47$	− 3.48270i	− 0.508003i	0.967204	−	0.254002i	$-0.0817469\pi$
$48$	0	0
$49$	5.18048	0.740069
$50$	0	0
$51$	−1.21509	−0.170147
$52$	0	0
$53$	2.51730i	0.345778i	0.984941	+	0.172889i	$0.0553102\pi$
$53$	2.51730i	0.345778i	−0.984941	+	0.172889i	$0.944690\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1.03461i	− 0.137037i
$58$	0	0
$59$	−7.86620	−1.02409	−0.512046	−	0.858958i	$-0.671112\pi$
$59$	−7.86620	−1.02409	−0.512046	+	0.858958i	$0.671112\pi$
$60$	0	0
$61$	−5.46479	−0.699695	−0.349848	−	0.936807i	$-0.613767\pi$
$61$	−5.46479	−0.699695	−0.349848	+	0.936807i	$0.613767\pi$
$62$	0	0
$63$	1.34889i	0.169945i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.41811i	0.173250i	0.996241	+	0.0866249i	$0.0276082\pi$
$67$	1.41811i	0.173250i	−0.996241	+	0.0866249i	$0.972392\pi$
$68$	0	0
$69$	−6.24970	−0.752376
$70$	0	0
$71$	3.79698	0.450619	0.225309	−	0.974287i	$-0.427661\pi$
$71$	3.79698	0.450619	0.225309	+	0.974287i	$0.427661\pi$
$72$	0	0
$73$	− 1.41811i	− 0.165977i	−0.996550	−	0.0829886i	$-0.973553\pi$
$73$	− 1.41811i	− 0.165977i	0.996550	−	0.0829886i	$-0.0264465\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.33682i	− 0.722148i
$78$	0	0
$79$	1.81952	0.204711	0.102356	−	0.994748i	$-0.467362\pi$
$79$	1.81952	0.204711	0.102356	+	0.994748i	$0.467362\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	13.2151i	1.45054i	0.688462	+	0.725272i	$0.258286\pi$
$83$	13.2151i	1.45054i	−0.688462	+	0.725272i	$0.741714\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.59859i	0.600232i
$88$	0	0
$89$	2.20302	0.233519	0.116760	−	0.993160i	$-0.462749\pi$
$89$	2.20302	0.233519	0.116760	+	0.993160i	$0.462749\pi$
$90$	0	0
$91$	−3.21509	−0.337033
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 6.69779i	− 0.680057i	−0.940415	−	0.340029i	$-0.889563\pi$
$97$	− 6.69779i	− 0.680057i	0.940415	−	0.340029i	$-0.110437\pi$
$98$	0	0
$99$	−4.69779	−0.472146
$100$	0	0
$101$	13.1280	1.30628	0.653141	−	0.757236i	$-0.273451\pi$
$101$	13.1280	1.30628	0.653141	+	0.757236i	$0.273451\pi$
$102$	0	0
$103$	− 16.4769i	− 1.62351i	−0.583995	−	0.811757i	$-0.698511\pi$
$103$	− 16.4769i	− 1.62351i	0.583995	−	0.811757i	$-0.301489\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.78491i	− 0.462574i	−0.972886	−	0.231287i	$-0.925706\pi$
$107$	− 4.78491i	− 0.462574i	0.972886	−	0.231287i	$-0.0742937\pi$
$108$	0	0
$109$	−0.947489	−0.0907530	−0.0453765	−	0.998970i	$-0.514449\pi$
$109$	−0.947489	−0.0907530	−0.0453765	+	0.998970i	$0.514449\pi$
$110$	0	0
$111$	5.41811	0.514264
$112$	0	0
$113$	17.1972i	1.61778i	0.587963	+	0.808888i	$0.299930\pi$
$113$	17.1972i	1.61778i	−0.587963	+	0.808888i	$0.700070\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.38350i	0.220355i
$118$	0	0
$119$	1.63903	0.150250
$120$	0	0
$121$	11.0692	1.00629
$122$	0	0
$123$	1.30221i	0.117416i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 2.16258i	− 0.191898i	−0.995386	−	0.0959490i	$-0.969411\pi$
$127$	− 2.16258i	− 0.191898i	0.995386	−	0.0959490i	$-0.0305886\pi$
$128$	0	0
$129$	−5.73240	−0.504709
$130$	0	0
$131$	10.9700	0.958455	0.479228	−	0.877691i	$-0.340917\pi$
$131$	10.9700	0.958455	0.479228	+	0.877691i	$0.340917\pi$
$132$	0	0
$133$	1.39558i	0.121012i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.67989i	0.741573i	0.928718	+	0.370786i	$0.120912\pi$
$137$	8.67989i	0.741573i	−0.928718	+	0.370786i	$0.879088\pi$
$138$	0	0
$139$	−4.76700	−0.404332	−0.202166	−	0.979351i	$-0.564798\pi$
$139$	−4.76700	−0.404332	−0.202166	+	0.979351i	$0.564798\pi$
$140$	0	0
$141$	3.48270	0.293296
$142$	0	0
$143$	− 11.1972i	− 0.936356i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	5.18048i	0.427279i
$148$	0	0
$149$	10.6978	0.876397	0.438198	−	0.898878i	$-0.355617\pi$
$149$	10.6978	0.876397	0.438198	+	0.898878i	$0.355617\pi$
$150$	0	0
$151$	5.01671	0.408254	0.204127	−	0.978944i	$-0.434565\pi$
$151$	5.01671	0.408254	0.204127	+	0.978944i	$0.434565\pi$
$152$	0	0
$153$	− 1.21509i	− 0.0982344i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	5.80161i	0.463019i	0.972833	+	0.231510i	$0.0743665\pi$
$157$	5.80161i	0.463019i	−0.972833	+	0.231510i	$0.925634\pi$
$158$	0	0
$159$	−2.51730	−0.199635
$160$	0	0
$161$	8.43018	0.664392
$162$	0	0
$163$	− 10.3143i	− 0.807877i	−0.914786	−	0.403939i	$-0.867641\pi$
$163$	− 10.3143i	− 0.807877i	0.914786	−	0.403939i	$-0.132359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.66318i	0.438230i	0.975699	+	0.219115i	$0.0703170\pi$
$167$	5.66318i	0.438230i	−0.975699	+	0.219115i	$0.929683\pi$
$168$	0	0
$169$	7.31892	0.562994
$170$	0	0
$171$	1.03461	0.0791185
$172$	0	0
$173$	− 11.6632i	− 0.886735i	−0.896340	−	0.443368i	$-0.853784\pi$
$173$	− 11.6632i	− 0.886735i	0.896340	−	0.443368i	$-0.146216\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 7.86620i	− 0.591260i
$178$	0	0
$179$	11.8258	0.883899	0.441949	−	0.897040i	$-0.354287\pi$
$179$	11.8258	0.883899	0.441949	+	0.897040i	$0.354287\pi$
$180$	0	0
$181$	−14.8604	−1.10456	−0.552281	−	0.833658i	$-0.686243\pi$
$181$	−14.8604	−1.10456	−0.552281	+	0.833658i	$0.686243\pi$
$182$	0	0
$183$	− 5.46479i	− 0.403969i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.70825i	0.417428i
$188$	0	0
$189$	−1.34889	−0.0981176
$190$	0	0
$191$	−5.93541	−0.429472	−0.214736	−	0.976672i	$-0.568889\pi$
$191$	−5.93541	−0.429472	−0.214736	+	0.976672i	$0.568889\pi$
$192$	0	0
$193$	− 3.19719i	− 0.230139i	−0.993357	−	0.115069i	$-0.963291\pi$
$193$	− 3.19719i	− 0.230139i	0.993357	−	0.115069i	$-0.0367090\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 18.7491i	− 1.33582i	−0.744243	−	0.667909i	$-0.767189\pi$
$197$	− 18.7491i	− 1.33582i	0.744243	−	0.667909i	$-0.232811\pi$
$198$	0	0
$199$	21.9129	1.55336	0.776681	−	0.629894i	$-0.216901\pi$
$199$	21.9129	1.55336	0.776681	+	0.629894i	$0.216901\pi$
$200$	0	0
$201$	−1.41811	−0.100026
$202$	0	0
$203$	− 7.55191i	− 0.530040i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 6.24970i	− 0.434384i
$208$	0	0
$209$	−4.86037	−0.336199
$210$	0	0
$211$	1.66318	0.114498	0.0572490	−	0.998360i	$-0.481767\pi$
$211$	1.66318	0.114498	0.0572490	+	0.998360i	$0.481767\pi$
$212$	0	0
$213$	3.79698i	0.260165i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 1.34889i	− 0.0915689i
$218$	0	0
$219$	1.41811	0.0958270
$220$	0	0
$221$	2.89618	0.194818
$222$	0	0
$223$	5.10382i	0.341777i	0.985290	+	0.170889i	$0.0546638\pi$
$223$	5.10382i	0.341777i	−0.985290	+	0.170889i	$0.945336\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 2.85412i	− 0.189435i	−0.995504	−	0.0947174i	$-0.969805\pi$
$227$	− 2.85412i	− 0.189435i	0.995504	−	0.0947174i	$-0.0301948\pi$
$228$	0	0
$229$	18.0934	1.19564	0.597822	−	0.801629i	$-0.296033\pi$
$229$	18.0934	1.19564	0.597822	+	0.801629i	$0.296033\pi$
$230$	0	0
$231$	6.33682	0.416932
$232$	0	0
$233$	− 9.01671i	− 0.590704i	−0.955389	−	0.295352i	$-0.904563\pi$
$233$	− 9.01671i	− 0.590704i	0.955389	−	0.295352i	$-0.0954369\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.81952i	0.118190i
$238$	0	0
$239$	−13.9642	−0.903269	−0.451634	−	0.892203i	$-0.649159\pi$
$239$	−13.9642	−0.903269	−0.451634	+	0.892203i	$0.649159\pi$
$240$	0	0
$241$	13.1972	0.850106	0.425053	−	0.905169i	$-0.360256\pi$
$241$	13.1972	0.850106	0.425053	+	0.905169i	$0.360256\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.46599i	0.156907i
$248$	0	0
$249$	−13.2151	−0.837472
$250$	0	0
$251$	−4.36097	−0.275262	−0.137631	−	0.990484i	$-0.543949\pi$
$251$	−4.36097	−0.275262	−0.137631	+	0.990484i	$0.543949\pi$
$252$	0	0
$253$	29.3598i	1.84583i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.6211i	− 0.724906i	−0.932002	−	0.362453i	$-0.881939\pi$
$257$	− 11.6211i	− 0.724906i	0.932002	−	0.362453i	$-0.118061\pi$
$258$	0	0
$259$	−7.30846	−0.454125
$260$	0	0
$261$	−5.59859	−0.346544
$262$	0	0
$263$	− 6.16258i	− 0.380001i	−0.981784	−	0.190001i	$-0.939151\pi$
$263$	− 6.16258i	− 0.380001i	0.981784	−	0.190001i	$-0.0608490\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.20302i	0.134823i
$268$	0	0
$269$	22.4590	1.36935	0.684674	−	0.728850i	$-0.259945\pi$
$269$	22.4590	1.36935	0.684674	+	0.728850i	$0.259945\pi$
$270$	0	0
$271$	−22.8362	−1.38720	−0.693601	−	0.720360i	$-0.743977\pi$
$271$	−22.8362	−1.38720	−0.693601	+	0.720360i	$0.743977\pi$
$272$	0	0
$273$	− 3.21509i	− 0.194586i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 14.1851i	− 0.852301i	−0.904652	−	0.426150i	$-0.859869\pi$
$277$	− 14.1851i	− 0.852301i	0.904652	−	0.426150i	$-0.140131\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	12.8362	0.765745	0.382872	−	0.923801i	$-0.374935\pi$
$281$	12.8362	0.765745	0.382872	+	0.923801i	$0.374935\pi$
$282$	0	0
$283$	5.70986i	0.339416i	0.985494	+	0.169708i	$0.0542825\pi$
$283$	5.70986i	0.339416i	−0.985494	+	0.169708i	$0.945718\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.75655i	− 0.103686i
$288$	0	0
$289$	15.5236	0.913150
$290$	0	0
$291$	6.69779	0.392631
$292$	0	0
$293$	8.01790i	0.468411i	0.972187	+	0.234205i	$0.0752488\pi$
$293$	8.01790i	0.468411i	−0.972187	+	0.234205i	$0.924751\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 4.69779i	− 0.272593i
$298$	0	0
$299$	14.8962	0.861468
$300$	0	0
$301$	7.73240	0.445688
$302$	0	0
$303$	13.1280i	0.754182i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	18.7445i	1.06980i	0.844914	+	0.534902i	$0.179651\pi$
$307$	18.7445i	1.06980i	−0.844914	+	0.534902i	$0.820349\pi$
$308$	0	0
$309$	16.4769	0.937336
$310$	0	0
$311$	−19.2260	−1.09020	−0.545102	−	0.838370i	$-0.683509\pi$
$311$	−19.2260	−1.09020	−0.545102	+	0.838370i	$0.683509\pi$
$312$	0	0
$313$	14.5219i	0.820828i	0.911899	+	0.410414i	$0.134616\pi$
$313$	14.5219i	0.820828i	−0.911899	+	0.410414i	$0.865384\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 0.378872i	− 0.0212796i	−0.999943	−	0.0106398i	$-0.996613\pi$
$317$	− 0.378872i	− 0.0212796i	0.999943	−	0.0106398i	$-0.00338681\pi$
$318$	0	0
$319$	26.3010	1.47257
$320$	0	0
$321$	4.78491	0.267067
$322$	0	0
$323$	− 1.25714i	− 0.0699494i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 0.947489i	− 0.0523963i
$328$	0	0
$329$	−4.69779	−0.258997
$330$	0	0
$331$	17.1459	0.942423	0.471211	−	0.882020i	$-0.343817\pi$
$331$	17.1459	0.942423	0.471211	+	0.882020i	$0.343817\pi$
$332$	0	0
$333$	5.41811i	0.296911i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.04668i	0.111490i	0.998445	+	0.0557450i	$0.0177534\pi$
$337$	2.04668i	0.111490i	−0.998445	+	0.0557450i	$0.982247\pi$
$338$	0	0
$339$	−17.1972	−0.934023
$340$	0	0
$341$	4.69779	0.254400
$342$	0	0
$343$	− 16.4302i	− 0.887147i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.26641i	0.497447i	0.968575	+	0.248723i	$0.0800110\pi$
$347$	9.26641i	0.497447i	−0.968575	+	0.248723i	$0.919989\pi$
$348$	0	0
$349$	3.41348	0.182719	0.0913597	−	0.995818i	$-0.470879\pi$
$349$	3.41348	0.182719	0.0913597	+	0.995818i	$0.470879\pi$
$350$	0	0
$351$	−2.38350	−0.127222
$352$	0	0
$353$	− 20.6799i	− 1.10068i	−0.834941	−	0.550340i	$-0.814498\pi$
$353$	− 20.6799i	− 1.10068i	0.834941	−	0.550340i	$-0.185502\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.63903i	0.0867467i
$358$	0	0
$359$	33.6227	1.77454	0.887270	−	0.461250i	$-0.152599\pi$
$359$	33.6227	1.77454	0.887270	+	0.461250i	$0.152599\pi$
$360$	0	0
$361$	−17.9296	−0.943662
$362$	0	0
$363$	11.0692i	0.580983i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	−1.30221	−0.0677904
$370$	0	0
$371$	3.39558	0.176290
$372$	0	0
$373$	35.2664i	1.82603i	0.407931	+	0.913013i	$0.366250\pi$
$373$	35.2664i	1.82603i	−0.407931	+	0.913013i	$0.633750\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.3443i	− 0.687265i
$378$	0	0
$379$	19.1038	0.981298	0.490649	−	0.871357i	$-0.336760\pi$
$379$	19.1038	0.981298	0.490649	+	0.871357i	$0.336760\pi$
$380$	0	0
$381$	2.16258	0.110792
$382$	0	0
$383$	20.0062i	1.02227i	0.859500	+	0.511136i	$0.170775\pi$
$383$	20.0062i	1.02227i	−0.859500	+	0.511136i	$0.829225\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 5.73240i	− 0.291394i
$388$	0	0
$389$	13.7036	0.694801	0.347400	−	0.937717i	$-0.387064\pi$
$389$	13.7036	0.694801	0.347400	+	0.937717i	$0.387064\pi$
$390$	0	0
$391$	−7.59396	−0.384043
$392$	0	0
$393$	10.9700i	0.553364i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	27.3598i	1.37315i	0.727060	+	0.686574i	$0.240886\pi$
$397$	27.3598i	1.37315i	−0.727060	+	0.686574i	$0.759114\pi$
$398$	0	0
$399$	−1.39558	−0.0698662
$400$	0	0
$401$	−9.79698	−0.489238	−0.244619	−	0.969619i	$-0.578663\pi$
$401$	−9.79698	−0.489238	−0.244619	+	0.969619i	$0.578663\pi$
$402$	0	0
$403$	− 2.38350i	− 0.118731i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 25.4531i	− 1.26167i
$408$	0	0
$409$	−16.2559	−0.803805	−0.401902	−	0.915683i	$-0.631651\pi$
$409$	−16.2559	−0.803805	−0.401902	+	0.915683i	$0.631651\pi$
$410$	0	0
$411$	−8.67989	−0.428147
$412$	0	0
$413$	10.6107i	0.522117i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.76700i	− 0.233441i
$418$	0	0
$419$	25.6920	1.25513	0.627567	−	0.778562i	$-0.284051\pi$
$419$	25.6920	1.25513	0.627567	+	0.778562i	$0.284051\pi$
$420$	0	0
$421$	−12.8541	−0.626472	−0.313236	−	0.949675i	$-0.601413\pi$
$421$	−12.8541	−0.626472	−0.313236	+	0.949675i	$0.601413\pi$
$422$	0	0
$423$	3.48270i	0.169334i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.37143i	0.356728i
$428$	0	0
$429$	11.1972	0.540605
$430$	0	0
$431$	36.8891	1.77689	0.888444	−	0.458985i	$-0.151787\pi$
$431$	36.8891	1.77689	0.888444	+	0.458985i	$0.151787\pi$
$432$	0	0
$433$	0.512673i	0.0246375i	0.999924	+	0.0123188i	$0.00392128\pi$
$433$	0.512673i	0.0246375i	−0.999924	+	0.0123188i	$0.996079\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.46599i	− 0.309310i
$438$	0	0
$439$	24.7219	1.17991	0.589957	−	0.807435i	$-0.299145\pi$
$439$	24.7219	1.17991	0.589957	+	0.807435i	$0.299145\pi$
$440$	0	0
$441$	−5.18048	−0.246690
$442$	0	0
$443$	35.4018i	1.68199i	0.541042	+	0.840996i	$0.318030\pi$
$443$	35.4018i	1.68199i	−0.541042	+	0.840996i	$0.681970\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	10.6978i	0.505988i
$448$	0	0
$449$	8.53984	0.403020	0.201510	−	0.979486i	$-0.435415\pi$
$449$	8.53984	0.403020	0.201510	+	0.979486i	$0.435415\pi$
$450$	0	0
$451$	6.11751	0.288063
$452$	0	0
$453$	5.01671i	0.235705i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	23.3131i	1.09054i	0.838260	+	0.545270i	$0.183573\pi$
$457$	23.3131i	1.09054i	−0.838260	+	0.545270i	$0.816427\pi$
$458$	0	0
$459$	1.21509	0.0567157
$460$	0	0
$461$	−28.7958	−1.34115	−0.670577	−	0.741840i	$-0.733953\pi$
$461$	−28.7958	−1.34115	−0.670577	+	0.741840i	$0.733953\pi$
$462$	0	0
$463$	− 18.7670i	− 0.872177i	−0.899904	−	0.436088i	$-0.856364\pi$
$463$	− 18.7670i	− 0.872177i	0.899904	−	0.436088i	$-0.143636\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.48270i	0.161160i	0.996748	+	0.0805800i	$0.0256772\pi$
$467$	3.48270i	0.161160i	−0.996748	+	0.0805800i	$0.974323\pi$
$468$	0	0
$469$	1.91288	0.0883286
$470$	0	0
$471$	−5.80161	−0.267324
$472$	0	0
$473$	26.9296i	1.23822i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.51730i	− 0.115259i
$478$	0	0
$479$	7.02998	0.321208	0.160604	−	0.987019i	$-0.448656\pi$
$479$	7.02998	0.321208	0.160604	+	0.987019i	$0.448656\pi$
$480$	0	0
$481$	−12.9141	−0.588831
$482$	0	0
$483$	8.43018i	0.383587i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	16.4543i	0.745617i	0.927908	+	0.372809i	$0.121605\pi$
$487$	16.4543i	0.745617i	−0.927908	+	0.372809i	$0.878395\pi$
$488$	0	0
$489$	10.3143	0.466428
$490$	0	0
$491$	−11.1972	−0.505322	−0.252661	−	0.967555i	$-0.581306\pi$
$491$	−11.1972	−0.505322	−0.252661	+	0.967555i	$0.581306\pi$
$492$	0	0
$493$	6.80281i	0.306383i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.12173i	− 0.229741i
$498$	0	0
$499$	2.74286	0.122787	0.0613935	−	0.998114i	$-0.480446\pi$
$499$	2.74286	0.122787	0.0613935	+	0.998114i	$0.480446\pi$
$500$	0	0
$501$	−5.66318	−0.253012
$502$	0	0
$503$	− 15.9821i	− 0.712606i	−0.934370	−	0.356303i	$-0.884037\pi$
$503$	− 15.9821i	− 0.712606i	0.934370	−	0.356303i	$-0.115963\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.31892i	0.325045i
$508$	0	0
$509$	−9.49357	−0.420795	−0.210398	−	0.977616i	$-0.567476\pi$
$509$	−9.49357	−0.420795	−0.210398	+	0.977616i	$0.567476\pi$
$510$	0	0
$511$	−1.91288	−0.0846209
$512$	0	0
$513$	1.03461i	0.0456791i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 16.3610i	− 0.719555i
$518$	0	0
$519$	11.6632	0.511957
$520$	0	0
$521$	6.83622	0.299500	0.149750	−	0.988724i	$-0.452153\pi$
$521$	6.83622	0.299500	0.149750	+	0.988724i	$0.452153\pi$
$522$	0	0
$523$	− 27.1972i	− 1.18925i	−0.804003	−	0.594625i	$-0.797301\pi$
$523$	− 27.1972i	− 1.18925i	0.804003	−	0.594625i	$-0.202699\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.21509i	0.0529303i
$528$	0	0
$529$	−16.0588	−0.698207
$530$	0	0
$531$	7.86620	0.341364
$532$	0	0
$533$	− 3.10382i	− 0.134442i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	11.8258i	0.510319i
$538$	0	0
$539$	24.3368	1.04826
$540$	0	0
$541$	41.6787	1.79191	0.895953	−	0.444148i	$-0.146494\pi$
$541$	41.6787	1.79191	0.895953	+	0.444148i	$0.146494\pi$
$542$	0	0
$543$	− 14.8604i	− 0.637720i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	19.0362i	0.813930i	0.913444	+	0.406965i	$0.133413\pi$
$547$	19.0362i	0.813930i	−0.913444	+	0.406965i	$0.866587\pi$
$548$	0	0
$549$	5.46479	0.233232
$550$	0	0
$551$	−5.79235	−0.246762
$552$	0	0
$553$	− 2.45433i	− 0.104369i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 36.2831i	− 1.53736i	−0.639631	−	0.768682i	$-0.720913\pi$
$557$	− 36.2831i	− 1.53736i	0.639631	−	0.768682i	$-0.279087\pi$
$558$	0	0
$559$	13.6632	0.577891
$560$	0	0
$561$	−5.70825	−0.241002
$562$	0	0
$563$	24.0755i	1.01466i	0.861752	+	0.507330i	$0.169367\pi$
$563$	24.0755i	1.01466i	−0.861752	+	0.507330i	$0.830633\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.34889i	− 0.0566482i
$568$	0	0
$569$	11.4360	0.479423	0.239711	−	0.970844i	$-0.422947\pi$
$569$	11.4360	0.479423	0.239711	+	0.970844i	$0.422947\pi$
$570$	0	0
$571$	24.2318	1.01407	0.507035	−	0.861926i	$-0.330742\pi$
$571$	24.2318	1.01407	0.507035	+	0.861926i	$0.330742\pi$
$572$	0	0
$573$	− 5.93541i	− 0.247955i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.53401i	− 0.0638616i	−0.999490	−	0.0319308i	$-0.989834\pi$
$577$	− 1.53401i	− 0.0638616i	0.999490	−	0.0319308i	$-0.0101656\pi$
$578$	0	0
$579$	3.19719	0.132871
$580$	0	0
$581$	17.8258	0.739537
$582$	0	0
$583$	11.8258i	0.489773i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.1626i	0.749650i	0.927096	+	0.374825i	$0.122297\pi$
$587$	18.1626i	0.749650i	−0.927096	+	0.374825i	$0.877703\pi$
$588$	0	0
$589$	−1.03461	−0.0426303
$590$	0	0
$591$	18.7491	0.771235
$592$	0	0
$593$	− 19.6274i	− 0.806000i	−0.915200	−	0.403000i	$-0.867968\pi$
$593$	− 19.6274i	− 0.806000i	0.915200	−	0.403000i	$-0.132032\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	21.9129i	0.896835i
$598$	0	0
$599$	0.609054	0.0248853	0.0124426	−	0.999923i	$-0.496039\pi$
$599$	0.609054	0.0248853	0.0124426	+	0.999923i	$0.496039\pi$
$600$	0	0
$601$	−23.6274	−0.963781	−0.481890	−	0.876232i	$-0.660050\pi$
$601$	−23.6274	−0.963781	−0.481890	+	0.876232i	$0.660050\pi$
$602$	0	0
$603$	− 1.41811i	− 0.0577499i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.1505i	1.67025i	0.550062	+	0.835124i	$0.314604\pi$
$607$	41.1505i	1.67025i	−0.550062	+	0.835124i	$0.685396\pi$
$608$	0	0
$609$	7.55191	0.306019
$610$	0	0
$611$	−8.30101	−0.335823
$612$	0	0
$613$	45.8966i	1.85375i	0.375375	+	0.926873i	$0.377514\pi$
$613$	45.8966i	1.85375i	−0.375375	+	0.926873i	$0.622486\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 46.8604i	− 1.88653i	−0.332044	−	0.943264i	$-0.607738\pi$
$617$	− 46.8604i	− 1.88653i	0.332044	−	0.943264i	$-0.392262\pi$
$618$	0	0
$619$	−11.8529	−0.476409	−0.238205	−	0.971215i	$-0.576559\pi$
$619$	−11.8529	−0.476409	−0.238205	+	0.971215i	$0.576559\pi$
$620$	0	0
$621$	6.24970	0.250792
$622$	0	0
$623$	− 2.97164i	− 0.119056i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 4.86037i	− 0.194104i
$628$	0	0
$629$	6.58351	0.262502
$630$	0	0
$631$	−17.2571	−0.686996	−0.343498	−	0.939153i	$-0.611612\pi$
$631$	−17.2571	−0.686996	−0.343498	+	0.939153i	$0.611612\pi$
$632$	0	0
$633$	1.66318i	0.0661055i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 12.3477i	− 0.489234i
$638$	0	0
$639$	−3.79698	−0.150206
$640$	0	0
$641$	−8.15795	−0.322220	−0.161110	−	0.986936i	$-0.551507\pi$
$641$	−8.15795	−0.322220	−0.161110	+	0.986936i	$0.551507\pi$
$642$	0	0
$643$	7.19719i	0.283829i	0.989879	+	0.141915i	$0.0453259\pi$
$643$	7.19719i	0.283829i	−0.989879	+	0.141915i	$0.954674\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.6873i	0.616733i	0.951268	+	0.308366i	$0.0997822\pi$
$647$	15.6873i	0.616733i	−0.951268	+	0.308366i	$0.900218\pi$
$648$	0	0
$649$	−36.9537	−1.45056
$650$	0	0
$651$	1.34889	0.0528673
$652$	0	0
$653$	− 37.8771i	− 1.48224i	−0.671370	−	0.741122i	$-0.734294\pi$
$653$	− 37.8771i	− 1.48224i	0.671370	−	0.741122i	$-0.265706\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.41811i	0.0553258i
$658$	0	0
$659$	46.3298	1.80475	0.902376	−	0.430949i	$-0.141821\pi$
$659$	46.3298	1.80475	0.902376	+	0.430949i	$0.141821\pi$
$660$	0	0
$661$	−9.19719	−0.357729	−0.178865	−	0.983874i	$-0.557242\pi$
$661$	−9.19719	−0.357729	−0.178865	+	0.983874i	$0.557242\pi$
$662$	0	0
$663$	2.89618i	0.112478i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	34.9895i	1.35480i
$668$	0	0
$669$	−5.10382	−0.197325
$670$	0	0
$671$	−25.6724	−0.991074
$672$	0	0
$673$	− 20.4169i	− 0.787014i	−0.919322	−	0.393507i	$-0.871262\pi$
$673$	− 20.4169i	− 0.787014i	0.919322	−	0.393507i	$-0.128738\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.7744i	− 0.606261i	−0.952949	−	0.303131i	$-0.901968\pi$
$677$	− 15.7744i	− 0.606261i	0.952949	−	0.303131i	$-0.0980319\pi$
$678$	0	0
$679$	−9.03461	−0.346716
$680$	0	0
$681$	2.85412	0.109370
$682$	0	0
$683$	11.4920i	0.439728i	0.975531	+	0.219864i	$0.0705613\pi$
$683$	11.4920i	0.439728i	−0.975531	+	0.219864i	$0.929439\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	18.0934i	0.690305i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	13.1638	0.500774	0.250387	−	0.968146i	$-0.419442\pi$
$691$	13.1638	0.500774	0.250387	+	0.968146i	$0.419442\pi$
$692$	0	0
$693$	6.33682i	0.240716i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.58231i	0.0599342i
$698$	0	0
$699$	9.01671	0.341043
$700$	0	0
$701$	−33.1372	−1.25158	−0.625788	−	0.779993i	$-0.715223\pi$
$701$	−33.1372	−1.25158	−0.625788	+	0.779993i	$0.715223\pi$
$702$	0	0
$703$	5.60562i	0.211420i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.7082i	− 0.665987i
$708$	0	0
$709$	−6.86037	−0.257647	−0.128823	−	0.991668i	$-0.541120\pi$
$709$	−6.86037	−0.257647	−0.128823	+	0.991668i	$0.541120\pi$
$710$	0	0
$711$	−1.81952	−0.0682372
$712$	0	0
$713$	6.24970i	0.234053i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 13.9642i	− 0.521502i
$718$	0	0
$719$	9.90663	0.369455	0.184728	−	0.982790i	$-0.440860\pi$
$719$	9.90663	0.369455	0.184728	+	0.982790i	$0.440860\pi$
$720$	0	0
$721$	−22.2256	−0.827723
$722$	0	0
$723$	13.1972i	0.490809i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 15.4423i	− 0.572722i	−0.958122	−	0.286361i	$-0.907554\pi$
$727$	− 15.4423i	− 0.572722i	0.958122	−	0.286361i	$-0.0924456\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−6.96539	−0.257624
$732$	0	0
$733$	40.9296i	1.51177i	0.654705	+	0.755884i	$0.272793\pi$
$733$	40.9296i	1.51177i	−0.654705	+	0.755884i	$0.727207\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	6.66198i	0.245397i
$738$	0	0
$739$	−50.4123	−1.85445	−0.927223	−	0.374510i	$-0.877811\pi$
$739$	−50.4123	−1.85445	−0.927223	+	0.374510i	$0.877811\pi$
$740$	0	0
$741$	−2.46599	−0.0905904
$742$	0	0
$743$	21.3505i	0.783274i	0.920120	+	0.391637i	$0.128091\pi$
$743$	21.3505i	0.783274i	−0.920120	+	0.391637i	$0.871909\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 13.2151i	− 0.483515i
$748$	0	0
$749$	−6.45433	−0.235836
$750$	0	0
$751$	32.1960	1.17485	0.587424	−	0.809279i	$-0.300142\pi$
$751$	32.1960	1.17485	0.587424	+	0.809279i	$0.300142\pi$
$752$	0	0
$753$	− 4.36097i	− 0.158923i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 28.2334i	− 1.02616i	−0.858341	−	0.513080i	$-0.828504\pi$
$757$	− 28.2334i	− 1.02616i	0.858341	−	0.513080i	$-0.171496\pi$
$758$	0	0
$759$	−29.3598	−1.06569
$760$	0	0
$761$	−33.9145	−1.22940	−0.614700	−	0.788761i	$-0.710723\pi$
$761$	−33.9145	−1.22940	−0.614700	+	0.788761i	$0.710723\pi$
$762$	0	0
$763$	1.27806i	0.0462690i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	18.7491i	0.676991i
$768$	0	0
$769$	−36.5990	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$769$	−36.5990	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$770$	0	0
$771$	11.6211	0.418525
$772$	0	0
$773$	− 33.2634i	− 1.19640i	−0.801346	−	0.598201i	$-0.795883\pi$
$773$	− 33.2634i	− 1.19640i	0.801346	−	0.598201i	$-0.204117\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 7.30846i	− 0.262189i
$778$	0	0
$779$	−1.34728	−0.0482713
$780$	0	0
$781$	17.8374	0.638273
$782$	0	0
$783$	− 5.59859i	− 0.200077i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16.2559i	− 0.579462i	−0.957108	−	0.289731i	$-0.906434\pi$
$787$	− 16.2559i	− 0.579462i	0.957108	−	0.289731i	$-0.0935658\pi$
$788$	0	0
$789$	6.16258	0.219394
$790$	0	0
$791$	23.1972	0.824797
$792$	0	0
$793$	13.0253i	0.462544i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 24.7157i	− 0.875475i	−0.899103	−	0.437737i	$-0.855780\pi$
$797$	− 24.7157i	− 0.875475i	0.899103	−	0.437737i	$-0.144220\pi$
$798$	0	0
$799$	4.23180	0.149710
$800$	0	0
$801$	−2.20302	−0.0778398
$802$	0	0
$803$	− 6.66198i	− 0.235096i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	22.4590i	0.790593i
$808$	0	0
$809$	−19.8211	−0.696874	−0.348437	−	0.937332i	$-0.613287\pi$
$809$	−19.8211	−0.696874	−0.348437	+	0.937332i	$0.613287\pi$
$810$	0	0
$811$	11.2330	0.394444	0.197222	−	0.980359i	$-0.436808\pi$
$811$	11.2330	0.394444	0.197222	+	0.980359i	$0.436808\pi$
$812$	0	0
$813$	− 22.8362i	− 0.800901i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 5.93078i	− 0.207492i
$818$	0	0
$819$	3.21509	0.112344
$820$	0	0
$821$	22.9700	0.801659	0.400830	−	0.916153i	$-0.368722\pi$
$821$	22.9700	0.801659	0.400830	+	0.916153i	$0.368722\pi$
$822$	0	0
$823$	0.697788i	0.0243234i	0.999926	+	0.0121617i	$0.00387128\pi$
$823$	0.697788i	0.0243234i	−0.999926	+	0.0121617i	$0.996129\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 30.0755i	− 1.04583i	−0.852386	−	0.522913i	$-0.824845\pi$
$827$	− 30.0755i	− 1.04583i	0.852386	−	0.522913i	$-0.175155\pi$
$828$	0	0
$829$	16.6620	0.578695	0.289347	−	0.957224i	$-0.406562\pi$
$829$	16.6620	0.578695	0.289347	+	0.957224i	$0.406562\pi$
$830$	0	0
$831$	14.1851	0.492076
$832$	0	0
$833$	6.29477i	0.218101i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	−16.9584	−0.585468	−0.292734	−	0.956194i	$-0.594565\pi$
$839$	−16.9584	−0.585468	−0.292734	+	0.956194i	$0.594565\pi$
$840$	0	0
$841$	2.34426	0.0808367
$842$	0	0
$843$	12.8362i	0.442103i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 14.9312i	− 0.513042i
$848$	0	0
$849$	−5.70986	−0.195962
$850$	0	0
$851$	33.8616	1.16076
$852$	0	0
$853$	− 5.43138i	− 0.185967i	−0.995668	−	0.0929835i	$-0.970360\pi$
$853$	− 5.43138i	− 0.185967i	0.995668	−	0.0929835i	$-0.0296404\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.6632i	0.398407i	0.979958	+	0.199203i	$0.0638354\pi$
$857$	11.6632i	0.398407i	−0.979958	+	0.199203i	$0.936165\pi$
$858$	0	0
$859$	−54.9779	−1.87582	−0.937911	−	0.346877i	$-0.887242\pi$
$859$	−54.9779	−1.87582	−0.937911	+	0.346877i	$0.887242\pi$
$860$	0	0
$861$	1.75655	0.0598629
$862$	0	0
$863$	20.4636i	0.696589i	0.937385	+	0.348294i	$0.113239\pi$
$863$	20.4636i	0.696589i	−0.937385	+	0.348294i	$0.886761\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	15.5236i	0.527207i
$868$	0	0
$869$	8.54770	0.289961
$870$	0	0
$871$	3.38007	0.114529
$872$	0	0
$873$	6.69779i	0.226686i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	33.1855i	1.12060i	0.828291	+	0.560298i	$0.189313\pi$
$877$	33.1855i	1.12060i	−0.828291	+	0.560298i	$0.810687\pi$
$878$	0	0
$879$	−8.01790	−0.270437
$880$	0	0
$881$	−5.30684	−0.178792	−0.0893960	−	0.995996i	$-0.528494\pi$
$881$	−5.30684	−0.178792	−0.0893960	+	0.995996i	$0.528494\pi$
$882$	0	0
$883$	29.6032i	0.996228i	0.867112	+	0.498114i	$0.165974\pi$
$883$	29.6032i	0.996228i	−0.867112	+	0.498114i	$0.834026\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 47.2727i	− 1.58726i	−0.608401	−	0.793630i	$-0.708189\pi$
$887$	− 47.2727i	− 1.58726i	0.608401	−	0.793630i	$-0.291811\pi$
$888$	0	0
$889$	−2.91709	−0.0978362
$890$	0	0
$891$	4.69779	0.157382
$892$	0	0
$893$	3.60323i	0.120577i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.8962i	0.497369i
$898$	0	0
$899$	5.59859	0.186724
$900$	0	0
$901$	−3.05876	−0.101902
$902$	0	0
$903$	7.73240i	0.257318i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.5010i	0.680725i	0.940294	+	0.340363i	$0.110550\pi$
$907$	20.5010i	0.680725i	−0.940294	+	0.340363i	$0.889450\pi$
$908$	0	0
$909$	−13.1280	−0.435427
$910$	0	0
$911$	7.62737	0.252706	0.126353	−	0.991985i	$-0.459673\pi$
$911$	7.62737	0.252706	0.126353	+	0.991985i	$0.459673\pi$
$912$	0	0
$913$	62.0817i	2.05460i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 14.7974i	− 0.488653i
$918$	0	0
$919$	6.92959	0.228586	0.114293	−	0.993447i	$-0.463540\pi$
$919$	6.92959	0.228586	0.114293	+	0.993447i	$0.463540\pi$
$920$	0	0
$921$	−18.7445	−0.617651
$922$	0	0
$923$	− 9.05012i	− 0.297888i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.4769i	0.541171i
$928$	0	0
$929$	39.6678	1.30146	0.650729	−	0.759310i	$-0.274463\pi$
$929$	39.6678	1.30146	0.650729	+	0.759310i	$0.274463\pi$
$930$	0	0
$931$	−5.35977	−0.175659
$932$	0	0
$933$	− 19.2260i	− 0.629430i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 31.6966i	− 1.03548i	−0.855537	−	0.517741i	$-0.826773\pi$
$937$	− 31.6966i	− 1.03548i	0.855537	−	0.517741i	$-0.173227\pi$
$938$	0	0
$939$	−14.5219	−0.473905
$940$	0	0
$941$	−36.8557	−1.20146	−0.600731	−	0.799451i	$-0.705124\pi$
$941$	−36.8557	−1.20146	−0.600731	+	0.799451i	$0.705124\pi$
$942$	0	0
$943$	8.13843i	0.265024i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	34.6107i	1.12470i	0.826901	+	0.562348i	$0.190102\pi$
$947$	34.6107i	1.12470i	−0.826901	+	0.562348i	$0.809898\pi$
$948$	0	0
$949$	−3.38007	−0.109722
$950$	0	0
$951$	0.378872	0.0122858
$952$	0	0
$953$	4.14468i	0.134259i	0.997744	+	0.0671297i	$0.0213841\pi$
$953$	4.14468i	0.134259i	−0.997744	+	0.0671297i	$0.978616\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26.3010i	0.850191i
$958$	0	0
$959$	11.7082	0.378079
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.78491i	0.154191i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	24.7553i	0.796078i	0.917368	+	0.398039i	$0.130309\pi$
$967$	24.7553i	0.796078i	−0.917368	+	0.398039i	$0.869691\pi$
$968$	0	0
$969$	1.25714	0.0403853
$970$	0	0
$971$	39.9930	1.28344	0.641718	−	0.766941i	$-0.278222\pi$
$971$	39.9930	1.28344	0.641718	+	0.766941i	$0.278222\pi$
$972$	0	0
$973$	6.43018i	0.206142i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 10.8604i	− 0.347454i	−0.984794	−	0.173727i	$-0.944419\pi$
$977$	− 10.8604i	− 0.347454i	0.984794	−	0.173727i	$-0.0555810\pi$
$978$	0	0
$979$	10.3493	0.330765
$980$	0	0
$981$	0.947489	0.0302510
$982$	0	0
$983$	5.03461i	0.160579i	0.996772	+	0.0802895i	$0.0255845\pi$
$983$	5.03461i	0.160579i	−0.996772	+	0.0802895i	$0.974416\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.69779i	− 0.149532i
$988$	0	0
$989$	−35.8258	−1.13919
$990$	0	0
$991$	55.0230	1.74786	0.873931	−	0.486050i	$-0.161563\pi$
$991$	55.0230	1.74786	0.873931	+	0.486050i	$0.161563\pi$
$992$	0	0
$993$	17.1459i	0.544108i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 5.40723i	− 0.171249i	−0.996327	−	0.0856244i	$-0.972711\pi$
$997$	− 5.40723i	− 0.171249i	0.996327	−	0.0856244i	$-0.0272885\pi$
$998$	0	0
$999$	−5.41811	−0.171421

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.q.3349.4		6
5.2	odd	4		1860.2.a.g.1.3	✓	3
5.3	odd	4		9300.2.a.u.1.1		3
5.4	even	2	inner	9300.2.g.q.3349.3		6
15.2	even	4		5580.2.a.j.1.3		3
20.7	even	4		7440.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.3	✓	3	5.2	odd	4
5580.2.a.j.1.3		3	15.2	even	4
7440.2.a.bn.1.1		3	20.7	even	4
9300.2.a.u.1.1		3	5.3	odd	4
9300.2.g.q.3349.3		6	5.4	even	2	inner
9300.2.g.q.3349.4		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.34889i q^{7} -1.00000 q^{9} +4.69779 q^{11} -2.38350i q^{13} +1.21509i q^{17} -1.03461 q^{19} +1.34889 q^{21} +6.24970i q^{23} -1.00000i q^{27} +5.59859 q^{29} +1.00000 q^{31} +4.69779i q^{33} -5.41811i q^{37} +2.38350 q^{39} +1.30221 q^{41} +5.73240i q^{43} -3.48270i q^{47} +5.18048 q^{49} -1.21509 q^{51} +2.51730i q^{53} -1.03461i q^{57} -7.86620 q^{59} -5.46479 q^{61} +1.34889i q^{63} +1.41811i q^{67} -6.24970 q^{69} +3.79698 q^{71} -1.41811i q^{73} -6.33682i q^{77} +1.81952 q^{79} +1.00000 q^{81} +13.2151i q^{83} +5.59859i q^{87} +2.20302 q^{89} -3.21509 q^{91} +1.00000i q^{93} -6.69779i q^{97} -4.69779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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